Quantitative dual-dye photometric method for determining dilution impact

ABSTRACT

The invention provides ways to determine the impact of diluting a solution wherein the diluting may be carried out for any of a variety of purposes. In one embodiment, the method enables accurate volume dispensation calculations independent of meniscus shape. In another embodiment, the method enables accurate determination of plate washing efficiency. In yet another embodiment, the method enables the accurate determination of dilution ratio over a plurality of dilution steps. The methods described may be carried out using one or more systems arranged to perform the steps. A kit of the invention includes instructions for carrying out the steps of the methods and, optionally, one or more solutions suitable for conducting photometric measurements.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application claims the priority benefit of U.S. provisionalpatent application Ser. No. 60/825,744, filed Sep. 15, 2006, entitled“QUANTITATIVE DUAL-DYE PHOTOMETRIC METHODS FOR DETERMINING VOLUMEDISPENSE ACCURACY AND/OR THE CORRESPONDING DILUTION RATIO FOR VARIEDLIQUIDS” of the present assignee. The present application also claimsthe priority benefit of U.S. provisional patent application Ser. No.60/940,766, filed May 30, 2007, entitled “QUANTITATIVE DUAL-DYEPHOTOMETRIC METHODS FOR DETERMINING VOLUME DISPENSE ACCURACY AND/OR THECORRESPONDING DILUTION RATIO FOR VARIED LIQUIDS” of the presentassignee. The entire contents of those prior applications areincorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to methods for determining the impact ofdiluting a solution. In particular, the present invention relates to amethod for determining a volume of a solution mixed with a diluent. Thepresent invention also relates to a method for determining theefficiency of removing one or more reagents from a vessel using a washsolution. The present invention further relates to a method fordetermining the dilution ratio for a solution under dilution.

2. Description of the Prior Art

Existing versions of the Multichannel Verification System (“MVS®”),which are commercially available from Artel, Inc. (“Artel”) ofWestbrook, Me. and are the subject of U.S. Pat. No. 6,741,365 and U.S.Pat. No. 7,187,455, include the use of equipment and aqueous samplesolutions to determine the accuracy of dispensing devices over specifiedvolume ranges. Additionally, an existing method which may be used tomeasure the volume of a non-aqueous sample solution created from anaqueous precursor is the subject of US Patent Application PublicationNo. 2007/0141709. The entire contents of U.S. Pat. No. 6,741,365, U.S.Pat. No. 7,187,455 and US Patent Application Publication No.2007/0141709 are incorporated herein by reference. Publication No2007/0141709 describes a general method for creating test solutions fromaqueous MVS® stock solutions and core calculation procedures correspondto those described in U.S. Pat. Nos. 6,741,365 and 7,187,455. Theaqueous-based solutions offered in the existing versions of the MVS®system meet the needs of many users, and the method for creating testsolutions from non-aqueous solvents described in Publication No.2007/0141709 also offers a test method needed by many users.

However, there are limitations to the MVS® system and to the method ofPublication No. 2007/0141709. For example, there are limitations to thetypes of solvents that may be used to create test solutions from MVS®stock solutions while still maintaining accurate calculations followingthe mathematical approaches described in U.S. Pat. Nos. 6,741,365 and7,187,455. Namely, any solvent that significantly alters the solutionmeniscus in the microtiter well can have a detrimental effect on theresults calculated by said approaches. The MVS® methods described inU.S. Pat. Nos. 6,741,365 and 7,187,455 are based on a generally flatmeniscus, a controlled or known solution chemistry that yields areproducible meniscus, and/or a correction factor which accounts forminor deviations in meniscus shape.

When an uncharacterized solvent included in a new test solution fromMVS® stock solutions (as described in Publication No. 2007/0141709) andthe existing MVS® methods (as described in U.S. Pat. Nos. 6,741,365 and7,187,455) are applied, there can be substantial errors, depending onhow curved the meniscus becomes. The existing MVS® methods rely uponknowledge of this meniscus curvature in order to accurately returnresults on the volume delivered to the microtiter well, and can only doso through application of a correction factor. Thus, for solvents thathave not been characterized for their affect on the solution meniscus,the results produced by the existing MVS® methods are less accurate thanthey could be. A method to cancel out the impact of variations inmeniscus would be useful in determining the volume of a solutiondispensed.

Various types of assays performed in life science and pharmaceuticallaboratories require wash steps to remove unwanted or used reagents froma reaction vessel. For example, Enzyme-linked Immunosorbent Assays(“ELISA assays”) require introduction of a tagged ligand. This taggedligand binds to the molecular entity of interest, if it is present, andcan be measured. The measurement is often a fluorometric, photometric orradiometric measurement, depending on the type of tag used. However,before the measurement step can occur, all unbound tagged-ligand must beremoved from the reaction vessel by a rinse, or wash, step with a washsolution, often consisting of buffered water.

For assays conducted in microtiter plates, wash steps are commonlyemployed to exchange the solution within the wells. These wash steps arecarried out using specialized equipment, called plate washers. Someexamples of plate washers include the Tecan Power Washer 384 (Tecan US,Durham, N.C.) and the Biotek ELx405 Microplate Washer (BioTekInstruments, Winooski, Vt.).

Plate washers typically operate by dispensing a wash solution into thewells of the microtiter plate, while at the same time removing solution.Thus, they incorporate a dispense tube, and an aspirate tube, oftenside-by-side. The solution is thus flowed into and out of the well at ahigh velocity in order to effectively flush out unwanted reagents. Thewash solution is dispensed into the wells by the dispense tip, which isconnected by tubing to a large volume reservoir (often greater than 1 L)containing clean wash solution. The aspirate tip is connected via tubingto a waste reservoir. The dispense tip is often inserted into the wellnear the bottom, thus introducing wash solution at the bottom of thewell. Conversely, the aspirate tip is inserted at a height near the top⅓ of the well height. Positioning the aspirate tip at this height forcesthe well contents to be pushed by the wash solution, flowing into thewell near the bottom, up towards the top of the well where it is removedby the aspiration tip. When the wash procedure is complete, the totalsolution height is equal to the height of the aspirate tip with respectto (or above) the plate bottom. This height thus determines the totalvolume of solution that will be left behind in the well.

For many microtiter plate-based assays incorporating a wash step, aquantitative understanding of the efficiency of washing is not needed.However, under some circumstances an understanding of how efficientlyreagents are being removed from the wells by the plate washer is needed.One way to measure washing efficiency is to determine the degree ofdilution that has occurred for the reagents in the wells. For suchtesting, the dilution testing scheme taught by existing methods can beused. However these existing methods require the use of a diluentsolution to be added to the wash reservoir. In many cases, adding such alarge volume of diluent is inconvenient and costly, especiallyconsidering a dead volume needed to fill the lines of 200-500 mL. Thus,an improved method for testing plate washing efficiency is desired.

Many test procedures carried out in life science and pharmaceuticallaboratories also require dilution-based volume transfer steps, such asdose response and detection limit assays, for example. For many of theseprocedures, quantitative measurements are collected and decisions aremade based upon an assumed, rather than a measured, dilution ratio.Often these assumptions are based upon a potentially misplaced trust inthe performance of automated liquid delivery equipment. Accuratelyknowing sample concentration is critical for properly interpreting theexperimental results, which can only be obtained if the experimentaldilution ratio is known and controlled. Thus, the ability to accuratelymeasure each dilution step in a dilution procedure having a plurality ofdilution steps is required for proper assay analysis.

What is needed is a method that enables accurate calculation of thevolume of a solution dispensed independent of meniscus. What is alsoneeded is an effective method to determine the efficiency of platewashing routines. Further, what is needed is a method to determinedilution accuracy in a dilution procedure including a plurality ofdilution steps.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide a method and relatedsystem to enable accurate calculation of the volume of a solutiondispensed independent of meniscus. It is also an object of the presentinvention to provide a system and related method to determine theefficiency of plate washing routines. Further, it is an object of thepresent invention to provide a system and related method to determinedilution accuracy in a dilution procedure including a plurality ofdilution steps.

The first method of the present invention is a dual-dye photometricmethod providing results that are independent of meniscus shape, and canthus be used for uncharacterized solvent types. Thus, this new method ismore widely applicable for volume testing of various solvents. In themethod, a quantity of diluent is first used to determine a pathlength oflight therethrough. It alone is used to account for meniscus shape in avessel. A solution under test is then added to the vessel and mixed withthe diluent. The solution can be of a solvent type which creates anunknown affect on the meniscus of the mixture but its volume calculationwill not be affected by the meniscus shape. Hence, this procedureprovides an option for testing a wider range of solvent types than theprocedure described in U.S. application Ser. No. 11/305,301.

The second method of the present invention also involves a dual-dyephotometric process providing results indicating plate washingefficiency. In this method, a solution contains a known concentration ofa first chromophore (sometimes referred to herein as a dye) at a veryhigh concentration that absorbs light at a first wavelength, but doesnot require another chromophore, and a diluent contains a knownconcentration of a second chromophore that absorbs light at a secondwavelength. The method involves dispensing the diluent into a vesselsuch that the volume of the diluent is larger than would be left behindby a plate washer. The plate washer is then used to aspirate and removethe volume of diluent to the point that the diluent height equals anaspirate tip height of the washer. The absorbance of the diluent is thenmeasured and used to determine a pathlength of light corresponding tothe diluent height in the vessel. That information is then used todetermine the dilutive effect of a plate washing instrument on thesolution with the first chromophore.

The third method is a dual-dye photometric method for accuratelydetermining each dilution step of a multi-step dilution process. Thismethod is based upon conducting the desired dilution procedure bydispensing a solution into a vessel and mixing it with a diluent,wherein the solution includes a first chromophore and a secondchromophore and the diluent includes only the second chromophore.Photometric measurements at made two different wavelengths. Themeasurements provided by this method are traceable to NIST for single ormultiple point dilutions, and cover a testable dilution range of up to1/2000. While this approach is similar to the standard dual-dye MVS®method described in U.S. Pat. No. 6,741,365, it provides a more preciseapproach for determining the degree of dilution experienced bydispensing one solution into a second solution, and allows for moreaccurate calculation of analyte concentrations throughout a multiplepoint dilution scheme.

In one embodiment of the present invention, a method includes the stepsof: (1) adding to the vessel a diluent including a known concentrationof a diluent chromophore which absorbs light at a second wavelength; (2)measuring the absorbance of the diluent chromophore at the secondwavelength; (3) adding a volume of the sample solution to the vessel,wherein the sample solution includes a known concentration of a samplesolution chromophore which absorbs light at a first wavelength, andwherein the sample solution does not include the diluent chromophore;(4) mixing the diluent and the sample solution in the vessel to producea mixture of the sample solution and the diluent; (5) measuring theabsorbance of the mixture of the sample solution and the diluent at thefirst wavelength and at the second wavelength; and (6) calculating thevolume of the sample solution added to the vessel based on the measuredabsorbances at the first wavelength and the second wavelength. The stepof calculating the volume of the sample solution also may include thesteps of first calculating the volume of the diluent added to the vesselbased on the absorbance per pathlength of the diluent at the secondwavelength, the pathlength of light through the diluent as determinedusing the measured absorbance of the diluent chromophore at the secondwavelength prior to adding the sample solution to the vessel, and thedimensions of the vessel. Further, the step of calculating the volume ofthe sample solution added to the vessel also may include a correctionfactor which accounts for a change in molar absorptivity.

In another embodiment of the invention, a method is provided thatincludes the steps of: (1) measuring in the sample solution theabsorbance of the first chromophore at the first wavelength and theabsorbance of the second chromophore at the second wavelength while thesample solution is contained in a first vessel of the plurality ofvessels; (2) transferring a target volume of the sample solution fromthe first vessel to a second vessel of the plurality of vessels; (3)mixing into the sample solution in the second vessel a target volume ofa diluent, wherein the diluent includes the second chromophore at aconcentration substantially equivalent to the concentration of thesecond chromophore in the sample solution; (4) measuring the absorbanceof the first chromophore at the first wavelength and the absorbance ofthe second chromophore at the second wavelength in the second vessel;and (5) calculating a dilution ratio for the sample solution containedin the second vessel, wherein the dilution ratio represents the extentto which the sample solution has been diluted by the diluent mixed intothe second vessel. The method may further include the steps of: (1)transferring a target volume of the mixture in the second vessel fromthe second vessel to a third vessel of the plurality of vessels; (2)mixing with the mixture in the third vessel a target volume of thediluent; (3) measuring the absorbance of the first chromophore at thefirst wavelength and the absorbance of the second chromophore at thesecond wavelength in the third vessel; and (4) calculating a dilutionratio for the mixture of the sample solution and the diluent containedin the third vessel, wherein the dilution ratio represents the extent towhich the mixture of the sample solution and the diluent has beendiluted by the diluent mixed into the third vessel.

The method may optionally include the steps of: (1) repeating X moretimes the steps of i) transferring the mixture of the sample solutionand the diluent, ii) mixing in the diluent, and iii) measuring theabsorbances, wherein X is ≧1, such that the last vessel of the pluralityof vessels with the mixture of the sample solution and the diluent andthe added diluent is vessel n and a preceding vessel is vessel m; and(2) calculating a dilution ratio for the mixture of the sample solutionand the diluent contained in vessel n, wherein the dilution ratiorepresents the extent to which the mixture of the sample solution andthe diluent has been diluted by the diluent mixed into vessel n.

In yet another embodiment of the present invention, a method is providedthat includes the steps of: (1) transferring a target volume of thesample solution from a source into a vessel; (2) mixing into the samplesolution in the vessel a target volume of the diluent, wherein thediluent includes the second chromophore at a concentration substantiallyequivalent to the known concentration of the second chromophore in thesample solution; (3) measuring the absorbance of the first chromophoreat the first wavelength and the absorbance of the second chromophore atthe second wavelength in the vessel; and (4) calculating a dilutionratio of the sample solution from the source, wherein the dilution ratiorepresents the extent to which the sample solution of the source hasbeen diluted by the diluent mixed into the vessel. This method mayinclude the steps of: (1) transferring a target volume of the mixture ofthe sample solution and the diluent from the vessel to a second vessel;(2) mixing a target volume of the diluent into the second vessel,wherein the diluent includes the second chromophore at a concentrationsubstantially equivalent to the known concentration of the secondchromophore in the sample solution; (3) measuring the absorbance of thefirst chromophore at the first wavelength and the absorbance of thesecond chromophore at the second wavelength in the second vessel; and(4) calculating a dilution ratio of the sample solution from the source,wherein the dilution ratio represents the extent to which the samplesolution of the source has been diluted by the diluent through allmixing steps.

The third embodiment of the present invention optionally may evenfurther include the optional steps of: (1) repeating X more times thesteps of i) transferring the mixture of the sample solution and thediluent, ii) adding the diluent with the known concentration of thesecond chromophore to the mixture of the sample solution and the diluentand iii) measuring the absorbances, wherein X is ≧1, such that anyvessel but the source is vessel m and the source is represented asvessel 0; and (2) calculating a dilution ratio for the sample solutionfrom the source, wherein the dilution ratio represents the extent towhich the sample solution of the source has been diluted by the diluentthrough all mixing steps.

In a fourth embodiment of the present invention, a method is providedthat includes the steps of: (1) adding to a first vessel of a first setof a plurality of vessels a target volume of a first sample solution,wherein the first sample solution includes a first known concentrationof the first chromophore and a known concentration of the secondchromophore; (2) adding to a first vessel of a second set of theplurality of vessels a target volume of a second sample solutionincluding a second known concentration of the first chromophore and theknown concentration of the second chromophore, wherein the second knownconcentration of the first chromophore of the second sample solution ishigher than the first known concentration of the first chromophore ofthe first sample solution, and wherein the target volume of the secondsample solution added to the first vessel of the second set issubstantially equivalent to the target volume of the first samplesolution added to the first vessel of the first set; (3) carrying out afirst dilution protocol step comprising: (i) mixing into the firstvessel of the first set a first target volume of a diluent, wherein thediluent includes the second chromophore at a concentration substantiallyequivalent to the known concentration of the second chromophore in thefirst sample solution and the second sample solution; (ii) mixing intothe first vessel of the second set the first target volume of thediluent; and (iii) measuring the absorbance of the first chromophore atthe first wavelength and the absorbance of the second chromophore at thesecond wavelength in the first vessel of the first set; (4) carrying outa second dilution protocol step comprising: (i) transferring a targetvolume of the mixture of the first sample solution and the diluent ofthe first vessel of the first set to a second vessel of the first set;(ii) mixing a second target volume of the diluent into the second vesselof the first set; (iii) transferring a target volume of the mixture ofthe second sample solution and the diluent of the first vessel of thesecond set to a second vessel of the second set; (iv) mixing the secondtarget volume of the diluent into the second vessel of the second set;and (v) measuring the absorbance of the first chromophore at the firstwavelength and the absorbance of the second chromophore at the secondwavelength in the second vessel of the second set; and (5) calculating adilution ratio of the dilution protocol based on the absorbancemeasurements made in steps 3.iii and 4.v, wherein the dilution ratiorepresents the extent of dilution occurring between the first dilutionprotocol step and the second dilution protocol step.

In a fifth embodiment of the present invention, a method is providedthat includes the steps of: (1) placing a diluent in a first vessel to alevel establishing a diluent height, wherein the diluent includes aknown concentration of a diluent chromophore that absorbs light at asecond wavelength, and wherein the first vessel has known dimensions;(2) measuring the absorbance of the diluent chromophore in the firstvessel at the second wavelength; (3) calculating the pathlength of lightthrough the diluent in the first vessel based on the measured absorbanceof the diluent chromophore and a known absorbance per pathlength of thediluent chromophore, wherein the pathlength of light through the diluentis equal to the diluent height; (4) adding to a second vessel a targetvolume of a sample solution, wherein the sample solution includes aknown concentration of a sample solution chromophore that absorbs lightat a first wavelength, wherein dimensions of the second vessel aresubstantially equivalent to the known dimensions of the first vessel;(5) adding a wash solution to the second vessel and removing at leastsome of a mixture of the wash solution and the sample solution from thesecond vessel to establish a mixture height in the second vessel,wherein the mixture height is substantially equivalent to the diluentheight; (6) measuring the absorbance of the sample solution chromophorein the second vessel at the first wavelength; and (7) calculating adilution ratio for the sample solution contained in the second vessel,wherein the dilution ratio represents the extent to which the samplesolution has been diluted by the wash solution added into the secondvessel. The method includes the option to carry out the step of addingwash solution to the second vessel and removing at least some of thewash solution and the sample solution from the second vessel a pluralityof times before the step of measuring the absorbance of the samplesolution chromophore. Alternatively, the steps of i) adding washsolution to the second vessel and removing at least some of the washsolution and the sample solution from the second vessel and ii)measuring the absorbance of the sample solution chromophore may becarried out a plurality of times in succession.

These and other features and advantages of the invention will beapparent upon review of the following detailed description, appendeddrawings and accompanying claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a table showing the approximate dilution range measurable byeach one of the five MVS® Sample Solutions of Example One, Example Twoand Example Three of the present invention described herein.

FIG. 2 is a table showing the experimental design which defines thecontents added to each column of a microtiter plate of Example One ofthe present invention described herein.

FIG. 3 is a table showing an experimental demonstration of Example Oneof the present invention described herein.

FIG. 4 is a table showing data obtained by carrying out Example Two ofthe present invention described herein.

FIG. 5 is a table showing two stepwise dilution calculations of ExampleThree of the present invention described herein.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS OF THE INVENTION

The present invention is a method and related system for determining theimpact of diluting a solution, either to accurately measure the volumeof the solution dispensed, or to accurately measure the amount ofdilution that has occurred. In a first system embodiment of the presentinvention includes the following components: (1) a microtiter platehaving a plurality of wells, wherein each well has a shape of knowngeometrical dimensions, (2) a microtiter plate reader for measuringabsorbance values of liquids dispensed into the wells of the microtiterplate, (3) sample solutions containing variable, but knownconcentrations of a first chromophore which absorbs light at a firstwavelength (herein referred to as “λ₁”), (4) a diluent solutioncontaining a known, fixed concentration of a second chromophore whichabsorbs light at a second wavelength (herein referred to as “λ₂”), and(5) a mixing device. It is to be understood, however, that the system isnot limited to having these components. Therefore, the system may havemore or fewer components, and other components may be substituted forany or all of these components of this first embodiment. Examples ofsuitable systems are described in the incorporated references.

A first primary method embodiment of the present invention includes thesteps of 1) adding a diluent to a well of a microtiter plate, whereinthe well has a shape of known geometrical dimensions; 2) measuring theabsorbance of the second chromophore in the well at the secondwavelength; 3) adding to the well a volume of sample solution, whereinthe sample solution includes a concentration of a sample solutionchromophore which absorbs light at a first wavelength; 4) mixing thediluent and the sample solution in the well to produce a mixture of thesample solution and the diluent; 5) measuring the absorbance of themixture of the sample solution and the diluent at the first wavelengthand at the second wavelength; and 6) calculating the volume of thesample solution added to the well.

The core mathematical model upon which the method of the presentinvention is based is the Beer-Lambert law. Simply stated, the lawclaims that when light is passed through a solution containing someconcentration of chromophore (i.e., dye), the amount of light absorbedby that dye solution is proportional to both the concentration of thechromophore and the interaction pathlength of the light with the dyesolution. In mathematical terms, the law is written as:A _(λ)=ε_(λ) lC  (1)where A_(λ) is the absorbance of the chromophore at a specificwavelength λ, ε_(λ) is a physical constant of the chromophore atwavelength λ known as the molar absorptivity, l is the pathlength oflight through the dye solution, and C is the molar concentration of thechromophore in the dye solution. This proportionality is most commonlyused to determine an unknown concentration of a chromophore in a dyesolution, where the molar absorptivity at the measurement wavelength isknown and the pathlength of light through the dye solution is known orfixed.

Alternatively, Equation (1) also states that the measured absorbance isproportional to the pathlength of light through the dye solution. Ifboth the molar absorptivity and concentration of the chromophore in adye solution are known, then Equation (1) can be used to determine anunknown pathlength traversed by a photometric light beam. To use theBeer-Lambert law in this way it becomes convenient to combine the molarabsorptivity (ε_(λ)) and concentration (C) terms:ε_(λ) C=a _(λ)  (2)where the new term a_(λ) represents the absorbance of the dye solutionper unit pathlength at the wavelength λ. Substitution of this new terminto Equation (1) gives:A _(λ) =a _(λ) l  (3)Equation (3) is the basic form of the Beer-Lambert law that is used forthe method of the present invention.

As noted, the first step of the first primary method involves adding avolume of diluent (V_(d)) into a well. The volume of diluent is chosenso that the total volume of sample solution (to be later added to thewell) and diluent does not exceed a defined working volume within thewell of the chosen plate type. For example, the defined working volumeused for each well of some 96-well microtiter plates is 200 μL, whereasthe working volume for each well of some 384-well plates is only 55 μL.

After dispensing the diluent, orbital mixing may be employed to ensuresufficient spreading of the diluent across the bottom surface of thewell, and flattening of the meniscus. This spreading may be achieved,for example, by using the mixing device of the apparatus of the presentinvention to agitate the mixture. If the sample solution under test doesnot allow for a flat meniscus, then the curvature of the meniscus mustbe characterized. Also, because an absorbance measurement will be madeof the dispensed diluent, spreading across the entire well bottom isrequired, otherwise a quantitative absorbance measurement will not beachieved. In practice this means that a lower limit must also be placedon the amount of the diluent volume used. For a 96-well plate, thislower limit is ˜50 μL.

After mixing, the absorbance at λ₂ (referred to as “A_(λ2)”) is measuredfor all wells containing diluent. Because the diluent contains a knownconcentration of the second chromophore, Equation (2) can be used tocharacterize the absorbance per pathlength of the diluent at λ₂(referred to as “a_(d)”). Thus, by incorporating the known a_(d) for thediluent and the measured absorbance A_(λ2) into Equation (3), thepathlength of light through the diluent (l_(d)) can be determined byusing the Equation:

$\begin{matrix}{l_{d} = {A_{\lambda\; 2}/a_{d}}} & (4)\end{matrix}$

By combining l_(d) with the known geometrical dimensions of themicrotiter wells, the volume of diluent in each well can be calculated.The exact equation used to determine the total diluent volume (V_(d)) isdependent upon the shape of the wells of the particular microtiter plateused. For commonly used round-well, flat-bottom plates, V_(d) is modeledas an inverted, truncated cone:

$\begin{matrix}{V_{d} = {{\pi\; l_{d}\frac{D^{2}}{4}} + {\pi\;{Dl}_{d}^{2}\frac{\tan\;\theta}{2}} + {\pi\; l_{d}^{3}\frac{\tan^{2}\theta}{3}}}} & (5)\end{matrix}$where D is the diameter of the bottom of the microtiter well, and θ isthe taper angle of the sidewall. Similar expressions (not given here)for determining V_(d) can be used when square-well microtiter plates areimplemented.

Once the diluent volume V_(d) has been determined, a sample solutionvolume, V_(s), is dispensed into the wells containing diluent. As withthe diluent, the sample solution possesses a characteristic absorbanceper pathlength at λ₁ (referred to as “a_(s)”) based on the knownconcentration of the first chromophore, as expressed in Equation (2). Itshould be noted again that in the first embodiment of the method, thesample solution contains only the first chromophore and none of thesecond chromophore.

After dispensing the sample solution into the wells containing diluent,the plate preferably is mixed until a homogenous mixture of sample anddiluent has been achieved, and absorbance measurements are then made atboth λ₁ and λ₂. The total volume of the combination of the samplesolution and diluent in any particular well is given by:V _(T) =V _(d) +V _(s)  (6)

Because each one of the sample solution and the diluent contains onlyone chromophore, and because the concentration of each chromophore isknown for each one of the sample solution and the diluent prior to theirmixing, Equation (3) can be used to express the measured absorbancevalues at each wavelength, λ₁ and λ₂, for the mixture:

$\begin{matrix}{A_{\lambda\; 1} = {a_{s}{l \cdot \left( \frac{V_{s}}{V_{T}} \right)}}} & (7) \\{A_{\lambda\; 2} = {a_{d}{l \cdot \left( \frac{V_{d}}{V_{T}} \right)}}} & (8)\end{matrix}$where l refers to the total pathlength of light through the homogenousmixture of sample solution and diluent, and the terms (V_(s)/V_(T)) and(V_(d)/V_(T)) correspond to the dilution ratios experienced by thesample solution and the diluent, respectively.

It is important to note that the total pathlength l may depend on eitheror both of the meniscus shape and the total solution volume V_(T).However, if a ratio of the absorbance values is calculated, the valuecalculated is independent of both l and V_(T). Dividing Equation (7) byEquation (8) results in removal of both l and V_(T) terms as expressedby:

$\begin{matrix}{\frac{A_{\lambda\; 1}}{A_{\lambda\; 2}} = {\frac{a_{s}}{a_{d}} \cdot \frac{V_{s}}{V_{d}}}} & (9)\end{matrix}$

Solving Equation (9) for the sample volume V_(s) gives:

$\begin{matrix}{V_{s} = {V_{d} \cdot \frac{a_{d}}{a_{s}} \cdot \frac{A_{\lambda\; 1}}{A_{\lambda\; 2}}}} & (10)\end{matrix}$

Because the diluent volume is determined in a separate step, it is theonly measurement whose meniscus shape is important to account for. Inother words, the diluent solution will need to have a relatively flatmeniscus when dispensed into the wells of a microtiter plate, or atleast have a controlled, reproducible meniscus that will allow forsufficient correction. Such a correction is needed because the meniscuscurvature causes the solution to no longer conform to the idealgeometrical shape of the well. In other words, the solution pathlengthmeasured photometrically no longer conforms to the height of the idealwell shape, but is either too short (for concave meniscus curvature) ortoo long (for convex meniscus curvature). A correction for such meniscuscurvature could be performed by determining the volume of diluent in thewells using a gravimetric approach, then determining the volume usingthe photometric approach. A correction factor can then be determined asthe difference between the gravimetric volume and the photometric volumeas a function of total solution pathlength in the well. Another possibleapproach for correcting for meniscus curvature would be to determine theactual radius of curvature and then calculate the enclosed volume of thegeometrical shape defined by the well dimensions and the meniscuscurvature.

In the first primary method embodiment of the invention, the samplesolution can be a solvent which creates an unknown affect on themeniscus of the mixture of the sample solution and the diluent. Whilethe meniscus curvature is accounted for when the diluent volume ismeasured, the sample volume calculation will not be affected by themeniscus shape as demonstrated by Equation (10).

While the first primary method embodiment is applicable to varioussample solvent types that may cause an uncharacterized change to themeniscus, it does not account for other interfering properties thatmight be created by the solvent of interest. For example, the samplesolution under test may contain some concentration of analyte thateither absorbs and/or scatters light at either or both λ₁ and λ₂. Undersuch circumstances, a blank solution containing the same concentrationof the interfering analyte as present in the total volume V_(T) may beprepared and used to collect a zero reading at both of thosewavelengths. When the blank solution is used in this manner, this zeroreading may then be subtracted from all raw absorbance measurementscollected at each wavelength before use in Equation (10). Those ofordinary skill in the art would recognize that such practice is commonlyused for photometric measurements.

In some cases, the first primary method embodiment may be affected bychemical or physical changes that a solvent of interest imparts on theeither or both of the chromophores of the mixture of sample solution andthe diluent. For example, some solvent types may cause precipitation ofeither or both chromophores. In this case, different chromophores thatare soluble in the solvent of interest would have to be used. In anotherexample, the solvent of interest may cause a change to the characterizedmolar absorptivity (ε) of either or both chromophores. Overcoming thisphysical spectral change would require characterization of the magnitudeof the imparted change, from which a correction could be made to themeasured absorbance values used in Equation (10).

As discussed below, a quantitative measurement of the efficiency ofplate washing can be achieved by determining the degree of dilution ofreagents. Existing dilution methods and the third primary embodiment ofthe method of the present invention described herein can be used toachieve this goal if the user is willing to consume several hundredmilliliters of diluent solution. The volume of solution required to fillthe dispense tubing is often 200-500 mL, and each wash step can requireapproximately 150-250 mL per plate. This is a significant volume thatmany users may not be willing to use, however.

A second primary method embodiment of the present invention allows theuser to avoid the need of having to fill the wash reservoir withdiluent. This second embodiment involves the use of two differentchromophore (dye) solutions. A first dye solution (e.g., a “red” dyesolution) contains a known concentration of a first chromophore at avery high concentration that absorbs light at a first wavelength (λ₁),but does not include any other chromophore. A second dye solution (e.g.,a “blue” dye solution) contains a known concentration of a secondchromophore that absorbs light at a second wavelength (λ₂). Whereas thefirst dye solution contains a high concentration of the firstchromophore, which has an immeasurable absorbance in its concentratedform, the second dye contains a known concentration of the secondchromophore at a spectrophotometrically measurable absorbance. Thesample solutions and diluent previously described for the first primarymethod embodiment have the requisite properties for this approach, andwill be used to further describe the second primary method embodiment ofthe invention. Because the concentration of each chromophore is knownfor each of the diluent and the sample solution, both the diluent andthe sample solution can be characterized by the absorbance perpathlength values as expressed in Equation (2). The absorbance perpathlength for the sample solution at the first chromophore (a_(s)), aswell as for the diluent at the second chromophore (a_(d)) are known andgiven by:a _(s)=ε_(λ1) ·C _(λ1)  (11)a _(d)=ε_(λ2) ·C _(λ2)  (12)where λ1 and λ2 are used to denote the first chromophore at the firstwavelength and the second chromophore at the second wavelength,respectively.

The second primary method embodiment involves dispensing the diluentusing a liquid handling apparatus (e.g., handheld pipette, automatedliquid handler, etc.), containing the known concentration of the secondchromophore into the wells of the microtiter plate. In the secondprimary method embodiment, the volume of the diluent is larger thanwould be left behind by the plate washer, meaning that the solutionheight is higher than the height of the aspirate tip of the platewasher. The microtiter plate is then loaded onto the plate washer andthe aspirate tips are inserted into the wells with the diluent. Becausethe diluent height is greater than the aspirate tip height, the aspiratetips are partially immersed into the diluent. The plate washer is thenused to aspirate and remove the volume of diluent to the point that thesolution height equals the aspirate tip height. At this point themicrotiter plate is removed and inserted into a plate reader, and theabsorbance of the diluent is measured at λ₂. From this step the totalsolution height can be measured. Equation (4) is used to measure theheight of the diluent in the well (l_(d)), which is equal to thesolution height that the plate washer will leave after a wash procedure.

A different microtiter plate is then filled with sample solution. Thissample solution contains a known concentration of the first chromophore,and is characterized by the absorbance per pathlength value shown inEquation (11). Although any type of liquid dispensing apparatus may beused to fill the wells, the volume dispensed into the wells should beequal to the volume of solution that would normally be present during anassay. This is the step that should mimic the assay being tested. Thus,if the washing efficiency during an ELISA assay is being tested, forexample, then the volume of sample placed in the wells should be thesame as the solution volume that would be present when the plate isinserted into the plate washer for washing. The plate filled with thedesired volume of sample solution is then inserted into the plate washerand the desired wash cycle is conducted. This wash cycle causes adilution of the sample solution, and leaves behind a total solutionvolume with a liquid height equal to the height that was previouslymeasured by the diluent test step (l_(d)=l_(solution)).

The diluted sample solution in the wells of the microtiter plate is thenmeasured photometrically with a plate reader. This measurement gives anabsorbance A_(s) at λ₁ for the first chromophore in the sample solution.The absorbance per pathlength for this diluted sample solution(a_(s,diluted)) can then be calculated by:

$\begin{matrix}{a_{s,{diluted}} = \frac{A_{s}}{l_{d}}} & (13)\end{matrix}$where the solution pathlength l_(d) is determined from the diluentfilled plate.

Once the a_(s,diluted) is determined, the plate washing efficiency canbe assessed by determining the extent of dilution experienced by thesample solution. This extent of dilution can be calculated as a dilutionratio, which is determined by comparing to the known absorbance perpathlength of the neat, concentrated sample solution (a_(s)) to theabsorbance per pathlength of the diluted sample solution (a_(s,diluted))left in the well after the wash cycle, as expressed by:

$\begin{matrix}{R_{WE} = \frac{a_{s}}{a_{s,{diluted}}}} & (14)\end{matrix}$where R_(WE) is the dilution ratio of the neat versus diluted samplesolution, and is taken as a measure of the degree of washing efficiencyof a plate washing cycle. (“Neat sample solution” is undiluted samplesolution).

Equation (14) provides a way to determine the efficiency of platewashing by comparing the measured dilution ratio against a desired levelof dilution. The ideal performance of a plate washer would result incomplete removal of unwanted components. However, adequate performancewould only require removal of unwanted reagents beyond somepre-determined threshold, which can be expressed in terms of a minimumdilution. The second embodiment of the method of the present inventionallows for testing the plate washer against such a dilution threshold.

In an alternative system embodiment of the present invention, the systemincludes: 1) a microtiter plate having wells for holding liquid volumes,2) a microtiter plate reader for measuring absorbance values ofsolutions dispensed into the wells of the microtiter plate, 3) samplesolutions containing variable, but known concentrations of a firstchromophore which absorbs light at λ₁, and a fixed, known concentrationof a second chromophore which absorbs light at λ₂, 4) a diluentcontaining a known, fixed concentration of a second chromophore whichabsorbs light at λ₂, and 5) a microtiter plate mixing apparatus. Itshould be noted that a significant difference between the standard MVS®approach described in U.S. Pat. No. 6,741,365 and the method of theinvention described herein is that the dimensions of the wells of themicrotiter plate are not needed for the present method. This is asignificant deviation from the standard MVS® approach as the platedimensions are used in the volume calculations of the standard MVS®approach.

In a third primary method embodiment of the present invention, stepsinclude: (1) transferring a target volume of the sample solution from asource into a vessel; (2) mixing into the sample solution in the vessela target volume of the diluent, wherein the diluent includes the secondchromophore at a concentration substantially equivalent to the knownconcentration of the second chromophore in the sample solution; (3)measuring the absorbance of the first chromophore at the firstwavelength and the absorbance of the second chromophore at the secondwavelength in the vessel; and (4) calculating a dilution ratio of thesample solution from the source, wherein the dilution ratio representsthe extent to which the sample solution of the source has been dilutedby the diluent mixed into the vessel. An object of the third primarymethod embodiment is to calculate the accuracy of each dilution step ascompared to the defined target.

Similar to the first primary method embodiment, the core mathematicalmodel upon which the third primary method embodiment is based is theBeer-Lambert law expressed in Equation (1). The concentration of eachchromophore is known for each solution, and is defined for the firstchromophore in the sample (a_(s)), and for the second chromophore in thediluent (a_(d)) by Equations (11) and (12). It should be noted that theconcentration of the second chromophore in the sample solution is fixedto equal the concentration of the second chromophore in the diluent.Thus, the absorbance per pathlength for all solutions, both samples anddiluent, at the second chromophore is fixed and can be expressed asa_(d). In practice this means that mixtures of any ratio of samplesolution to diluent will result in the same concentration of the secondchromophore, but a varying concentration of the first chromophore. Thisin essence allows the second chromophore to be used as an internalstandard.

The first step of the third primary method embodiment entails dispensingsample solution volume V_(s1) into well 1, followed by removal of volumeV_(s2) from well 1 and dispensing it into well 2. The net volume ofsample solution left in well 1 (V₁) is given by:V ₁ =V _(s1) −V _(s2)  (15)

Because the concentration of the first and second chromophores in well 1are known, the depth of liquid (l₁) in well 1 can be determined bymeasuring the absorbance of the second chromophore and using Equation(3) above. More directly expressed, the depth of liquid is given by:

$\begin{matrix}{l_{1} = \frac{A_{1,{\lambda\; 2}}}{a_{d}}} & (16)\end{matrix}$where A_(1,λ2) denotes the measured absorbance in well 1 at the secondwavelength λ₂, and a_(d) is the absorbance per pathlength of the secondchromophore.

The Beer's law expression for the absorbance of the first chromophore inwell 1 is given by:A _(1,λ1)=ε_(λ1) ·C _(1,λ1) ·l ₁  (17)where C_(1,λ1) is used to indicate the concentration in well 1 of thefirst chromophore. Substituting Equations (11) and (16) into (17) givesthe following:

$\begin{matrix}{A_{1,{\lambda\; 1}} = {A_{1,{\lambda\; 2}} \cdot \frac{a_{s}}{a_{d}}}} & (18)\end{matrix}$

A volume of diluent (V_(d2)) is then added to the sample solution volume(V_(s2)) already present in well 2. The sample solution and diluent aremixed until homogenous and a volume V_(s3) of the mixture of samplesolution and diluent is then removed and dispensed into well 3. The netvolume in well 2 is given by:V ₂ =V _(s2) +V _(d2) −V _(s3)  (19)

Because the concentration of the second chromophore is fixed for allsample solutions and diluent, it is known for well 2 and the solutiondepth can be determined by the measured absorbance of the secondchromophore at λ₂, as given by:

$\begin{matrix}{l_{2} = \frac{A_{2,{\lambda\; 2}}}{a_{d}}} & (20)\end{matrix}$

The concentration of the first chromophore in well 2 has been dilutedand is given by:

$\begin{matrix}{C_{2,{\lambda\; 1}} = {C_{1,{\lambda\; 1}} \cdot \left( \frac{V_{s\; 2}}{V_{s\; 2} + V_{d\; 2}} \right)}} & (21)\end{matrix}$where

(V_(s 2)/V_(s 2) + V_(d 2))represents the dilution factor of the first chromophore in going fromwell 1 to well 2.

The Beer's law expression for the absorbance of the first chromophore inwell 2 is given by:A _(2,λ1)=ε_(λ1) ·C _(2,λ1) ·l ₂  (22)

Substitution of Equations (11), (20) and (21) into (22) gives thefollowing reduced expression:

$\begin{matrix}{A_{2,{\lambda\; 1}} = {A_{2,{\lambda\; 2}} \cdot \frac{a_{s}}{a_{d}} \cdot \left( \frac{V_{s\; 2}}{V_{s\; 2} + V_{d\; 2}} \right)}} & (23)\end{matrix}$

A first a volume of diluent (V_(d3)) is then added to the mixture ofsample solution and diluent volume (V_(s3)) present in well 3. Themixture of sample solution and diluent are mixed and a volume V_(s4) isthen removed and dispensed into well 4. As occurred for well 2, theexpressions for the net volume and solution depth for well 3 are givenas:V ₃ =V _(s3) +V _(d3) −V _(s4)  (24)

$\begin{matrix}{l_{3} = \frac{A_{3,{\lambda\; 2}}}{a_{d}}} & (25)\end{matrix}$

The concentration of the first chromophore in well 3 has been diluted bya ratio of

(V_(s 3)/V_(s 3) + V_(d 3))and is given by:

$\begin{matrix}{C_{3,{\lambda\; 1}} = {C_{2,{\lambda\; 1}} \cdot \left( \frac{V_{s\; 3}}{V_{s\; 3} + V_{d\; 3}} \right)}} & (26)\end{matrix}$

Substitution of the expression for the concentration of the firstchromophore in well 2 from Equation (21) into Equation (26) gives:

$\begin{matrix}{C_{3,{\lambda\; 1}} = {C_{1,{\lambda\; 1}} \cdot \left( \frac{V_{s\; 2}}{V_{s\; 2} + V_{d\; 2}} \right) \cdot \left( \frac{V_{s\; 3}}{V_{s\; 3} + V_{d\; 3}} \right)}} & (27)\end{matrix}$

The Beer's law expression for the absorbance of the first chromophore inwell 3 is given as:A _(3,λ1)=ε_(λ1) ·C _(3λ1) ·l ₃  (28)

Substitution of Equations (11), (26) and (27) into (28) gives thefollowing reduced expression:

$\begin{matrix}{A_{3,{\lambda\; 1}} = {A_{3,{\lambda\; 2}} \cdot \frac{a_{s}}{a_{d}} \cdot \left( \frac{V_{s\; 2}}{V_{s\; 2} + V_{d\; 2}} \right) \cdot \left( \frac{V_{s\; 3}}{V_{s\; 3} + V_{d\; 3}} \right)}} & (29)\end{matrix}$

The steps described above can be followed to develop expressions forcontinued analysis of well 4 and beyond.

An object of this analysis is to describe a method for calculating thedilution ratio of each step. The dilution ratio in going from well 1 towell 2 is defined as:

$\begin{matrix}{R_{12} \equiv \frac{C_{1,{\lambda\; 1}}}{C_{2,{\lambda\; 1}}}} & (30)\end{matrix}$

Inserting the expression for the concentration of the first chromophorein well 2 from Equation (21) into Equation (30) gives the followingexpression:

$\begin{matrix}{{R_{12} \equiv \frac{C_{1,\;{\lambda\; 1}}}{C_{1,{\lambda\; 1}} \cdot \left( \frac{V_{S\; 2}}{V_{S\; 2} + V_{D\; 2}} \right)}} = \left( \frac{V_{S\; 2} + V_{D\; 2}}{V_{S\; 2}} \right)} & (31)\end{matrix}$

The Beer's law expression for the absorbance ratio of the firstchromophore in well 1 compared to well 2 is given through dividingEquation (18) by Equation (23):

$\begin{matrix}{\frac{A_{1,{\lambda\; 1}}}{A_{2,{\lambda\; 1}}} = \frac{\frac{a_{s}}{a_{d}} \cdot A_{1,{\lambda\; 2}}}{\frac{a_{s}}{a_{d}} \cdot {A_{2,{\lambda\; 2}}\left( \frac{V_{S\; 2}}{V_{S\; 2} + V_{D\; 2}} \right)}}} & (32)\end{matrix}$

Substituting Equation (31) and simplifying gives:

$\begin{matrix}{\frac{A_{1,{\lambda\; 1}}}{A_{2,{\lambda\; 1}}} = {\frac{A_{1,{\lambda\; 2}}}{A_{2,{\lambda\; 2}}} \cdot R_{12}}} & (33)\end{matrix}$

Solving for the dilution ratio gives:

$\begin{matrix}{R_{12} = {\frac{A_{1,{\lambda\; 1}}}{A_{2,{\lambda\; 1}}} \cdot \frac{A_{2,{\lambda\; 2}}}{A_{1,{\lambda\; 2}}}}} & (34)\end{matrix}$

Likewise, in going from well 2 to well 3, the dilution ratio is:

$\begin{matrix}{R_{23} = {\frac{A_{2,{\lambda\; 1}}}{A_{3,{\lambda\; 1}}} \cdot \frac{A_{3,{\lambda 2}}}{A_{2,{\lambda 2}}}}} & (35)\end{matrix}$

A more general expression for dilution steps is given as:

$\begin{matrix}{R_{mn} = {\frac{A_{m,{\lambda 1}}}{A_{n,\;{\lambda 1}}} \cdot \frac{A_{n,{\lambda\; 2}}}{A_{m,{\lambda 2}}}}} & (36)\end{matrix}$

The analysis described above demonstrates that the dilution ratios foreach well can be determined by measuring the absorbance ratios of thefirst and second chromophores for each well. It is important to notethat neither the dye concentration nor the well dimensions are requiredfor this analysis. As demonstrated by Equation (36), the only quantitiesneeded to calculate a dilution ratio in going from one well to anotherare the measured absorbance values for both the first and secondchromophores. Also, this analysis is independent of meniscus shape, solong as the meniscus curvature is not significantly different betweenwell m and well n. The only limitation for the stepwise analysis ofdilution testing described by Equation (36) is that the absorbancevalues for both chromophores have to be in a measurable absorbance rangefor the spectrophotometer used. In other words, for this dilutioncalculation to be valid, the absorbance of both the first and secondchromophores have to be in the linear, Beer's law absorbance range inboth well m and well n.

As discussed, Equation (36) allows for calculating the degree ofdilution between any two dilution steps in a series of dilutions of aprotocol, but only if the absorbance values for both chromophores is ina measurable range in both wells. This requirement can pose asignificant limitation for commonly used dilution protocols which covergreater than a 1,000 fold dilution between the start and end wells. Theabove approach provides a method for overcoming such a limitation, solong as the starting concentrations for both chromophores are known. Forexample, assume that the dilution ratio in going from well 1 to well 3is to be determined. Such a dilution is expressed as:

$\begin{matrix}{R_{13} = {{R_{12} \cdot R_{23}} = {\left( {\frac{A_{1,{\lambda 1}}}{A_{2,{\lambda 1}}} \cdot \frac{A_{2,{\lambda 2}}}{A_{1,{\lambda 2}}}} \right) \cdot \left( {\frac{A_{2,{\lambda 1}}}{A_{3,{\lambda 1}}} \cdot \frac{A_{3,{\lambda 2}}}{A_{2,{\lambda 2}}}} \right)}}} & (37)\end{matrix}$

Simplifying Equation (37) demonstrates that the dilution step betweenwell 1 and well 3 can be calculated directly, without any measurementsof well 2:

$\begin{matrix}{R_{13} = {\frac{A_{1,{\lambda 1}}}{A_{3,{\lambda 1}}} \cdot \frac{A_{3,{\lambda 2}}}{A_{1,{\lambda 2}}}}} & (38)\end{matrix}$

Because the contents of well 1 are defined, Equation (18) can beincorporated into Equation (38) to give:

$\begin{matrix}{R_{13} = {\frac{a_{s}}{a_{d}} \cdot \frac{A_{3,{\lambda 2}}}{A_{3,{\lambda 1}}}}} & (39)\end{matrix}$This can be more generally stated as:

$\begin{matrix}{R_{0m} = {\frac{a_{s}}{a_{d}} \cdot \frac{A_{m,{\lambda 2}}}{A_{m,{\lambda 1}}}}} & (40)\end{matrix}$where R_(0m) refers to the dilution experienced by a starting solutionwith known concentrations of both the first and second chromophores thatis dispensed into well m which contains diluent. This dilutioncalculation can be applied between two wells where the first wellcontains a known concentration of both chromophores, and the second wellcontains a dilution that results in measurable absorbance values forboth the first and second chromophores.

For the case where the concentrated solution is contained in well 1,Equation (40) is used to calculate the dilution ratio in going from well1 to well m, where well 1 contains neat sample solution. For very largedilution steps, it may be required to start with a highly concentratedchromophore solution which is beyond the measurable absorbance range ofthe plate reader in order to have a measurable concentration ofchromophore in well m. Such a dilution step is still measurable so longas the concentrations of both the first and second chromophores areknown for this solution. Thus, Equation (40) provides the means tocalculate accuracy of very large dilution steps.

Equations (36) and (40) demonstrate how the third primary methodembodiment is used to measure two different types of dilution step.Equation (36) can be used to measure smaller dilution steps that resultin measurable absorbance values in both wells m and n, where thesolution has been diluted from well m into well n. Equation (40) is fordetermining large step dilutions where the starting solution is neatsample solution. In the case of Equation (40), the absorbance obtainedfrom well m is used, along with the absorbance per pathlength values forthe neat sample solution.

Another type of dilution protocol step to be considered in use of thethird primary method is one that entails a multi-step procedure wherethe absorbance of either well m or well n is immeasurable. Equation (40)allows for calculating the dilution ratio for the first step of such aprotocol by starting with a sample solution with a known absorbance perpathlength value for both dyes. By selecting a sample solution with anappropriate concentration, the first dilution step will produce adiluted solution with a measurable absorbance at both 520 nm and 730 nm,and Equation (40) will accurately calculate the dilution that hasoccurred. However, suppose that the dilution procedure then required asecond large dilution step that resulted in a red dye absorbance thatwas too small to measure. The way in which this second dilution step isassessed is by repeating the dilution method in a different microtiterplate, but starting with a more concentrated sample solution. Data fromboth plates can then be used to determine the dilution step that hasoccurred in going from well m into well n, as defined by:

$\begin{matrix}{R_{m_{i}n_{j}} = \frac{\left( \frac{a_{s_{j}}}{a_{d_{j}}} \right) \cdot \left( \frac{A_{n_{j},\lambda_{2}}}{A_{n_{j},\lambda_{1}}} \right)}{\left( \frac{a_{s_{l}}}{a_{d_{i}}} \right) \cdot \left( \frac{A_{m_{i},{\lambda 2}}}{A_{m_{i},\lambda_{1}}} \right)}} & (41)\end{matrix}$where R_(m) _(i) _(n) _(j) refers to the dilution that has occurredbetween well m and well n for the dilution protocol that was performed,where the data for well m is taken from plate i, and the data for well nis taken from plate j. To use Equation (41), the same dilution protocolis conducted for two different plates: 1) plate i using a samplesolution that results in a measurable absorbance in well m, and 2) platej using a different, more concentrated sample solution that results in ameasurable absorbance in well n. It is instructive to note that Equation(41) is the quotient of the dilution calculations from Equation (40) forwell m in plate i and well n in plate j.

By using the dual-dye, dual-wavelength method described above, dilutionratios for all steps of a dilution scheme can be measured usingEquations (36), (40) and (41). Not all dilutions will be within themeasurable range of the plate reader, however. To overcome thislimitation, the initial concentrations of the chromophores in solutionwill need to characterized and known. Also, for multi-step dilutions,one starting solution may not be capable of covering all dilution steps.For such a scenario, different starting solutions with differentstarting concentrations will need to be dispensed following the multiplesteps of the dilution protocol. While one starting solution may notresult in measurable absorbance values beyond the first few steps,another starting solution may not provide measurable absorbance valuesuntil after multiple steps in the dilution scheme have already beenmade. Thus, if the dilution method is followed using different startingsolutions, the accuracy of each step can be determined.

To fully understand the accuracy of each step in a multi-step dilutionprotocol, one must consider the measurable absorbance range of thesample solutions used. For example, consider a four-step serial dilutionprotocol where each step consists of a 1:4 dilution. Assuming a startingconcentration of a first chromophore that yields an absorbance of 2.5,which is within the measurable absorbance range of mostspectrophotometers, Equation (1) shows that the absorbance will bereduced by a factor of 1/64 after the third dilution, which correspondsto an absorbance value of 0.039. While such a small absorbance ismeasurable by many spectrophotometers, the noise component of such asmall value can be significant. Thus, the sample solution used in thiscase provides measurable quantities only for the first two dilutionsteps of this protocol. Now assume that the protocol is repeated for amore concentrated sample solution whose first chromophore concentrationwould yield an equivalent absorbance of 75, if such a measurement wereachievable by a spectrophotometer. The absorbance of the firstchromophore would still be beyond measurement after the first and seconddilution steps. However, the absorbance of the third (1:64) and fourth(1:256) dilution steps would be 1.17 and 0.29, respectively. Thus, theprotocol is conducted two separate times, one using the sample solutionwith an initial absorbance of 2.5 for the first chromophore, the secondusing the more concentrated sample solution with an initial absorbanceof 75 for the first chromophore. By using these two different samplesolutions with different concentrations of the first chromophore, andrepeating the dilution protocol for each solution, all four steps of the1:4 serial dilution protocol can be assessed using Equations (36), (40)and (41), assuming that the first chromophore absorbance per pathlengthvalues (a_(s)) can be determined for each starting sample solution.

The third primary method embodiment of the present invention is morespecifically described with reference to three Examples; however, it isnot to be construed as being limited thereto.

Example ONE

The process of Example One was carried out for the purpose ofdetermining the dispense accuracy of a liquid handling device, whichspecifically was a Rainin, 20-200 μL LTS multichannel pipette.

The process of Example One included the use of a plurality of MVS®Diluent and Sample Solutions (i.e., “Range A”, “Range B”, “Range C”,“Range D” and “Range E” sample Solutions), which are commerciallyavailable from Artel, Inc. of Westbrook, Me. The MVS® Sample Solutionsincluded a common, fixed blue dye (chromophore) concentration and avariable red dye (chromophore) concentration. MVS® Diluent contained thesame blue dye concentration as in the MVS® Sample Solutions. Theconcentrations of red and blue dyes in each of the Sample Solutions andthe Diluent are known and well controlled through a rigorous qualitycontrol process.

The MVS® Diluent and Sample Solutions are characterized by theabsorbance per pathlength values for their red and blue dyes, as definedin Equations (11) and (12). Considering the known absorbance perpathlength of red dye (a_(s)) in each MVS® Sample Solution, the table ofFIG. 1 reports the approximate dilution range that may be achieved foreach Solution. These dilution ranges are based upon maintaining ameasurable absorbance of a mixture of a Sample Solution and Diluent ofbetween 0.3 and 2.4. The a_(s) in each MVS® Sample Solution wasdetermined through a large volume gravimetric dilution process. A largevolume dilution was gravimetrically made for each MVS® Sample Solutionusing a 5-place analytical balance (Mettler-Toledo, AX205), and theabsorbance of this dilution was measured with a horizontal beam UV-Visspectrophotometer (Varian, Cary 5000) in a cuvette of known pathlength.By making an accurate gravimetric dilution having an absorbance whichwas within the measurable range of the spectrophotometer, an equivalenta_(s) value was determined for highly concentrated MVS® SampleSolutions.

200 μL of undiluted (neat) MVS® Sample Solution was dispensed from themultichannel pipette under test into each well in column 1 of an SBSstandard 96-well microtiter plate (Costar, 3631). 100 μL of the SampleSolution was aspirated from each well of column 1 using the multichannelpipette and each one of these aliquots of Sample Solutions was dispensedinto a separate well of column 2, which contained 100 μL of MVS® Diluentsolution. Each of the contents of the wells in column 2 were mixed byaspirating and dispensing 100 μL three times. A 67 μL volume wasaspirated from each well of column 2 and each one of these mixtures ofDiluent and Sample Solutions was dispensed into column 3, whichcontained 133 μL of MVS® Diluent, and the mixing step was repeated forthe contents of all wells of column 3. This process continued forcolumns 3 thru 7, with the exception that a different volume of themixture of Diluent and Sample Solution was dispensed into the wells ofeach column, as defined in the table of FIG. 2. The absorbance of eachsolution-filled well was measured at 520 nm and 730 nm using amicrotiter plate reader (Bio-Tek Instruments, ELx800nb).

This protocol was followed for all five MVS® Sample Solutions, meaningfive separate microtiter plates were prepared, one for each MVS® SampleSolution. All five MVS® Sample Solutions were used to ensure ameasurable absorbance for each step of the defined protocol, thusallowing for testing the individual steps of the multi-step dilutionprotocol defined in the table of FIG. 2.

The table of FIG. 3 summarizes the performance of the multichannelpipette when conducting the defined dilution protocol, and consists ofdata compiled from all five plates used in this experimental protocol.The inaccuracies of the average dilution ratio are reported, which arebased upon the average dilution of the n=8 wells in each column. Themeasured dilution inaccuracy was calculated versus the target dilutionusing the formula: Inaccuracy=(Measured−Target)/Target. While only theaverage inaccuracies are reported herein, the inaccuracy on a tip-to-tipbasis may also be determined, which would allow for the analysis ofchannel-to-channel repeatability for the multi-channel pipette.

The ‘Inaccuracy of Stepwise Dilution’ data of FIG. 3 shows theuncertainty of the transfer of sample from one column to the next. Thisinaccuracy calculation was based upon the dilution ratio calculatedusing Equation (34). The ‘Inaccuracy of Total Dilution’ of FIG. 3represents the error associated with the overall dilution ratio for aspecific column with respect to column 1, and was calculated using thedilution ratio from Equation (38).

Example One shows that by using the third primary method embodiment ofthe present invention, it is possible to determine the accuracy of eachstep of a variable step dilution protocol. Additionally, when using theMVS® Sample Solutions, the data collected indicate that the assumed1:2000 endpoint dilution ratio can be achieved. In fact, if theacceptable absorbance range for the Range E solution is lowered to 0.19,the measurable dilution ratio extends to almost 1:3000. While anabsorbance of 0.19 is measurable for most spectrophotometers, the effectof noise in that measurement should not be ignored.

While a fluorescence-based method could allow for dilution testingbeyond a 1:10,000 ratio, the range covered by the MVS® Sample Solutionsshould allow for testing many commonly performed dilution assays. If adye with a higher molar absorptivity is used, the testable dilutionrange could be expanded significantly. For example, many heme porphyrinshave a molar absorptivity (ε) of >100,000 M⁻¹cm⁻¹, which is five timesgreater than the ε for the red dye used in the MVS® Sample Solution.Using such a dye would clearly increase the testable dilution range tonearly 1:10,000 for this absorbance-based approach.

It should be noted that the results of this approach are independent ofseveral factors, including: i) well size, ii) well shape, iii) wellmaterial, and iv) the interaction effects between the solution and theplate material, such as meniscus and air pockets, unless the light beamis obstructed. It should also be noted that this process is highlydependent upon the thoroughness of mixing, as are all dilution basedmethods. The best performance of any assay based upon dilution schema orupon photometric or fluorometric measurements requires complete mixing,which should be independently assessed.

Example TWO

The process of Example Two included the use of the plurality of MVS®Diluent and Sample Solutions described in Example One and is summarizedin representative form in the table of FIG. 4. Generally, 200 μL ofundiluted (neat) MVS® Sample Solution was dispensed into each well ofcolumn 1 of a 96-well plate (i.e., into 8 wells total). 100 μL of theSample Solution was aspirated from each well of column 1 using themultichannel pipette of Example One and dispensed into the correspondingwells of column 2, each of which contained 100 μL of Diluent solution.The contents of the wells in column two were then mixed by aspiratingand dispensing 100 μL three times. 100 μL of the contents of each wellof column 2 were aspirated and dispensed into the corresponding wells ofcolumn 3, each of which contained 100 μL of Diluent solution. The mixingstep was then repeated for column 3. These steps were repeated acrossthe plate resulting in 1:2 dilution steps for each well. The 100 μLsample aspirated from column 12 (i.e., the last column of the platefilled) was discarded to waste. The plate was then mixed on an orbitalshaker at 1300 RPM for one minute. The same protocol was followed usingnew plates for all five MVS® Sample Solutions. Since these five MVS®Sample Solutions increase in concentration (i.e., from “Range A” through“Range E”) to allow low volume measurement with a maintained absorbanceresponse, all dilution steps could be tested and measured.

Based on the known concentrations of MVS® Sample Solutions, thefollowing dilution ranges were possible while maintaining an absorbanceresponse between 0.3 and 2.4 absorbance units: 1) Range A=1 to 1/4dilution, 2) Range B=1/4 to 1/20 dilution, 3) Range C=1/20 to 1/100dilution, 4) Range D=1/100 to 1/400 dilution, and 5) Range E=1/400 to1/2000 dilution.

All five Sample Solutions were analyzed starting from their neat form byperforming 1:2 dilutions across an entire 96-well microtiter plate.Individual plates were filled for each solution. The appropriate SampleSolution was used for each dilution to ensure the proper absorbanceresponse based on the calculated dilution range of the solution, asdescribed above. Averages are reported herein. However, inaccuracy on atip-to-tip basis may be determined, which allows for the analysis ofchannel-to-channel repeatability reported as ‘% CV tip-to-tip’ in thetable of FIG. 4. ‘Inaccuracy from initial’ represents the errorassociated with the overall dilution ratio relating the initial dispensein column 1 and a specific column as calculated by Equation (38).‘Inaccuracy of transfer’ data shows the uncertainty of the transfer ofsample from one column to the next as calculated by Equation (34).Example Two shows that it is possible to determine the accuracy of eachstep of a serial dilution protocol up to a 1:2000 endpoint ratio byusing the third primary method embodiment of the present invention.

Example THREE

Example Three demonstrates the use of the two stepwise dilutioncalculations presented in Equations (36) and (41) and is represented bythe table of FIG. 5. The method of Example Three was not carried out inthe same fashion as in the previous two Examples, wherein dilutions wereproduced from one well to the next in a microtiter plate. Instead, tominimize errors associated with making dilutions, gravimetric dilutionswere made of MVS Range C Sample Solution (“Range C”). Three dilutionswere gravimetrically made. The first dilution was a 1:20 fold dilutionof Range C, made by weighing a desired amount of Range C into a bottle,followed by weighing the desired amount of MVS Diluent. The componentswere mixed. After mixing, a 1:2 dilution of the mixed sample wasperformed gravimetrically into a new bottle, and the contents weremixed. A third 1:2 dilution was performed into yet a new bottle. Thisresulted in three dilutions of Range C; 1:20, 1:40, 1:80.

Each of the three dilutions of Range C were dispensed into all wells ofthree 96-well microtiter plates using a multichannel pipette such thateach dilution of Range C was included in a separate plate. The totalvolume of each dilution dispensed into each well was 200 μL. Afterfilling all wells, each of the three plates was mixed using an orbitalmixer; the purpose for mixing was to evenly spread the solutionmeniscus. After mixing, the absorbance of all wells in each plate wasmeasured at 520 nm and 730 nm.

The dilution step for each plate was calculated using Equations (36) and(41), as presented in the table of FIG. 5. Unlike the previous Examples,the stepwise dilutions calculated with Equation (36) did not involvecomparing the measured absorbance from one column of wells in one plateto another column of wells in the same plate. Instead, the averageabsorbance of all 96 wells in each plate was calculated, and thestepwise dilutions were calculated from one plate to the next.

Some important items to note from this example are: 1) The first 1:20dilution step of Range C is not measurable using Equation (36) becausethe absorbance of the Range C Sample Solution is too high to be directlymeasured, as demonstrated in FIG. 1. However, this 1:20 dilution step ismeasurable by Equation (41), as expected. 2) A direct comparison of thestepwise dilution approaches presented in Equations (36) and (41) can bemade, and should be equivalent for the experiment performed herein. Thedata indicates that these two calculations are equivalent, demonstratingthe validity of both approaches.

The present invention is also embodied in a kit in one or more forms.The kit of the present invention may include any one or more componentsof the system of the invention. For example, the kit may include thesample solution, the diluent, the first chromophore, the secondchromophore, the diluent chromophore, the sample solution chromophore,the microtiter plate, and/or the vessel. When the kit includes thediluent, the first chromophore, the second chromophore, the diluentchromophore, the sample and/or the solution chromophore, and alsoincludes the microtiter plate and/or the vessel, the included diluent,the first chromophore, the second chromophore, the diluent chromophore,and/or the sample solution chromophore may be contained in the micotiterplate and/or the vessel.

Further, the kit may include instructions for carrying out one or moreof the methods described herein using one or more of the systemsdescribed herein or other systems suitable to carry out the steps of themethods described. The kit of the present invention also may furtherinclude computer-executable software stored on a computer-readablemedium, the computer-executable software being capable of performing anyone or more of the calculations steps described herein and/or to effectautomated performance of one or more steps described.

For example, the computer-executable software may includecomputer-readable signals tangibly embodied on the computer-readablemedium, where such signals define instructions for processing dataobtained by carrying out the method of the invention. Such instructionsmay be written in any of a plurality of programming languages, forexample, Java, XML, Visual Basic, C, or C++, Fortran, Pascal, Eiffel,BASIC, COBOL, and the like, or any of a variety of combinations thereof.The computer-readable medium on which such instructions preferablyreside is to be compatible with the central processing unit of thecomputing system. Further, the steps of processing the data may beperformed in alternative orders, in parallel and serially.

It is to be understood that various modifications may be made to themethod, system and/or the kit without departing from the spirit andscope of the invention. For example, the steps of the method may beperformed in differing order, one or more steps may be omitted, and oneor more steps may be replaced with alternative forms thereof.Accordingly, other embodiments are within the scope of the claimsappended hereto.

1. A method of determining the dilution achieved in carrying out adilution protocol using a sample solution and a diluent, wherein thesample solution includes a first chromophore that absorbs light at afirst wavelength and a known concentration of a second chromophore thatabsorbs light at a second wavelength, the method comprising the stepsof: a. adding to a first vessel of a first set of a plurality of vesselsa target volume of a first sample solution, wherein the first samplesolution includes a first known concentration of the first chromophoreand a known concentration of the second chromophore, wherein each of thefirst chromophore and the second chromophore has an absorbance maximum,the absorbance maximum of the first chromophore minimally overlappingwith the absorbance maximum of the second chromophore; b. adding to afirst vessel of a second set of the plurality of vessels a target volumeof a second sample solution including a second known concentration ofthe first chromophore and the known concentration of the secondchromophore, wherein the second known concentration of the firstchromophore of the second sample solution is higher than the first knownconcentration of the first chromophore of the first sample solution, andwherein the target volume of the second sample solution added to thefirst vessel of the second set is substantially equivalent to the targetvolume of the first sample solution added to the first vessel of thefirst set; c. carrying out a first dilution protocol step comprising: i.mixing into the first vessel of the first set a first target volume of adiluent, wherein the diluent includes the second chromophore at aconcentration substantially equivalent to the known concentration of thesecond chromophore in the first sample solution and the second samplesolution; ii. mixing into the first vessel of the second set the firsttarget volume of the diluent; and iii. measuring the absorbance of thefirst chromophore at the first wavelength and the absorbance of thesecond chromophore at the second wavelength in the first vessel of thefirst set; d. carrying out a second dilution protocol step comprising:i. transferring a target volume of the mixture of the first samplesolution and the diluent of the first vessel of the first set to asecond vessel of the first set; ii. mixing a second target volume of thediluent into the second vessel of the first set; iii. transferring atarget volume of the mixture of the second sample solution and thediluent of the first vessel of the second set to a second vessel of thesecond set; iv. mixing the second target volume of the diluent into thesecond vessel of the second set; and v. measuring the absorbance of thefirst chromophore at the first wavelength and the absorbance of thesecond chromophore at the second wavelength in the second vessel of thesecond set; and e. calculating a dilution ratio of the dilution protocolbased on the absorbance measurements made in steps c.iii and d.v,wherein the dilution ratio represents the extent of dilution occurringbetween the first dilution protocol step and the second dilutionprotocol step.
 2. The method of claim 1 wherein the calculation of thedilution ratio involves using the equation:$R_{12}^{\prime} = \frac{\left( \frac{a_{s_{2}}}{a_{d_{2}}} \right) \cdot \left( \frac{A_{2,\lambda_{2}}}{A_{2,\lambda_{1}}} \right)}{\left( \frac{a_{s_{1}}}{a_{d_{1}}} \right) \cdot \left( \frac{A_{1,\lambda_{2}}}{A_{1,\lambda_{1}}} \right)}$where (R′₁₂) represents the dilution ratio achieved after completing thesecond dilution protocol step, (as₁) is the absorbance per pathlength ofthe first chromophore at the first wavelength in the first samplesolution, (as₂) is the absorbance per pathlength of the firstchromophore at the first wavelength in the second sample solution, (ad₁)is the absorbance per pathlength of the second chromophore at the secondwavelength in the diluent of the first set; (ad₂) is the absorbance perpathlength of the second chromophore at the second wavelength in thediluent of the second set, (A_(1,λ1)) is the absorbance of the firstchromophore at the first wavelength measured in the first vessel of thefirst set, (A_(1,λ2)) is the absorbance of the second chromophore at thesecond wavelength measured in the first vessel of the first set,(A_(2,λ1)) is the absorbance of the first chromophore at the firstwavelength measured in the second vessel of the second set, and(A_(2,λ2)) is the absorbance of the second chromophore at the secondwavelength measured in the second vessel of the second set.
 3. Themethod of claim 1 further comprising the steps of: a. repeating X moretimes the dilution protocol steps of i) transferring the sample solutionand the diluent mixture to subsequent vessels for each of the first andsecond sets; ii) mixing a target volume of the diluent into thesubsequent vessels and iii) measuring the absorbances of the first andsecond chromophores, wherein X is ≧1, such that the last dilutionprotocol step is dilution protocol step n and a preceding dilutionprotocol step is dilution protocol step m, and wherein vessels of thefirst set and the second set containing the mixture of the samplesolution and the diluent after completing dilution protocol step m arerepresented as vessel m and vessels of the first and second setcontaining the mixture of the sample solution and the diluent aftercompleting dilution protocol step n are represented as vessel n; and b.calculating a dilution ratio of the dilution protocol based on theabsorbance measurements made after completing dilution protocol step mand dilution protocol step n, wherein the dilution ratio represents theextent of dilution occurring between dilution protocol step m anddilution protocol step n.
 4. The method of claim 3 wherein thecalculation of the dilution ratio involves using the equation:$R_{m_{i}n_{j}} = \frac{\left( \frac{a_{s_{j}}}{a_{d_{j}}} \right) \cdot \left( \frac{A_{n_{j},\lambda_{2}}}{A_{n_{j},\lambda_{1}}} \right)}{\left( \frac{a_{s_{i}}}{a_{d_{i}}} \right) \cdot \left( \frac{A_{m_{i},\lambda_{2}}}{A_{m_{i},\lambda_{1}}} \right)}$where (Rm_(i)n_(j)) represents the dilution ratio achieved aftercompleting dilution protocol step n, (as_(i)) is the absorbance perpathlength of the first chromophore at the first wavelength in the firstsample solution, (as_(j)) is the absorbance per pathlength of the firstchromophore at the first wavelength in the second sample solution,(ad_(i)) is the absorbance per pathlength of the second chromophore atthe second wavelength in the diluent of the first set; (ad_(j)) is theabsorbance per pathlength of the second chromophore at the secondwavelength in the diluent of the second set, (A_(mi,λ1)) is theabsorbance of the first chromophore at the first wavelength measured invessel m of the first set, (A_(mi,λ2)) is the absorbance of the secondchromophore at the second wavelength measured in vessel m of the firstset, (A_(nj,λ1)) is the absorbance of the first chromophore at the firstwavelength measured in vessel n of the second set, and (A_(nj,λ2)) isthe absorbance of the second chromophore at the second wavelengthmeasured in vessel n of the second set.
 5. The method of claim 1 whereinthe first target volume of the diluent and the second target volume ofthe diluent are not equal.